The OF algorithms discussed above were posed as calculus of variations problems. In this section, we provide an alternative interpretation of these methods using the results from linear smoothing theory. In particular, we develop stochastic state-space models where the effect of the brightness constraint equation is specified through an observation equation, and the effect of regularization is specified through a state-space equation. This viewpoint allows the computation of a priori error measures. Further, by making the underlying motion model explicit, it allows the development of new OF methods using alternate smoothness conditions. It is particularly important to our plans for 3D implementation where the property of incompressibility can be used to advantage.
Recall from Section 2.2 that HSOF uses a smoothness condition
wherein the magnitude of the gradients of 
and 
is penalized.
Suter [21] proposed using smoothness conditions on the first
order differential invariants of 
, namely its
divergence and curl.
We note that, via the Helmholtz decomposition of
into solenoidal and irrotational
components [22], this is equivalent to imposing smoothness on the
scalar and vector potentials of the field.
These conditions are physically more meaningful as they can
incorporate known physical properties
of the velocity field, such as an incompressibility condition.
Using these smoothness conditions, we get the following
variational problems [23] :
the first order DIV-CURL spline
or the second order DIV-CURL spline
or the second order DIV-CURL spline
where 
stands for the gradient of 
.
Suter showed that (12) is equivalent to the standard
Horn and Schunck functional when 
(see also [21], [24]).
In 3D, the motion of the heart is incompressible to a good
approximation. This can be easily modeled by the above
smoothness splines by picking 
. This ability
to tune the regularization according to the physical properties
of the flow field makes (12) and (13)
attractive for use in OF.
As pointed above, it is advantageous to consider an alternative viewpoint for these variational formulations. We consider linear stochastic state-space systems of the following form:
where the velocity is assumed to be defined on a regular domain 
with boundary 
. Here
, 
,
and the subscript 
implies a restriction of
the variable to the boundary .
The observation equations are as follows:
where
and 
.
The solution to this linear smoothing problem can be found by using
the theory of complimentary processes, and Adams et al. [25]
have shown that the linear minimum mean squared error estimate
is given by
where 
is the adjoint of operator 
, and
, is the
normal derivative of 
. Here, the symbol 
means
Kronecker product.
We have shown in [17] that by suitably choosing the
parameters in (14) and (15),
the solution in (16) can be made to match the
variational solution of (12) and (13).
Thus, we get a stochastic system which incorporates the regularization
through the action of the operator , and which incorporates a noisy
version of the BCE through the observation equation.
The advantage of this approach is that a priori error
measures - such as the estimation error covariance matrix - can be
computed. This approach also makes the a priori information
about the velocity field explicit, which helps both in
understanding the performance in any given situation and also in the
development of new OF algorithms where the actual physics of the
problem might be exploited.
Rougee et al. [26], used a similar approach to develop a state-space model for HSOF. However, we have shown in [17] that their model is over-determined, giving a non-physical model which cannot be used in simulations. In the same paper, we have developed well-posed state-space models for (12) and (13). Finally, we have shown that a well-posed state space model for HSOF is obtained as a special case from our model. Our formulation can be used to simulate velocity fields that are tuned to the model. This was demonstrated theoretically and experimentally in [16]. In a recent work, we have extended these methods to 3D [27].
